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Long time asymptotics of the Korteweg-de Vries equation. (English) Zbl 0619.35084

The \(t\to \infty\) behaviour of the solution u(x,t) of the KdV equation with a step-like initial condition: \(u(x,0)=v(x)\lim_{x\to -\infty} v(x)=-1\), \(\lim_{t\to \infty} v(x)=0\), is studied. It is known [E. Ya. Khruslov, Mat. Sb., Nov. Ser. 99(141), 261-281 (1976; Zbl 0332.35022)] that at \(t\to \infty\) a wave train of solitons is emitted. The result of the paper is the following asymptotic lower bound for the number N(t) of solitons separated at time t: for arbitrary positive \(\epsilon\), \(N(t)\geq t^{1/4-\epsilon}\) as \(t\to \infty\).
Reviewer: G.Nenciu

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
76B25 Solitary waves for incompressible inviscid fluids

Citations:

Zbl 0332.35022
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References:

[1] E. Ja. Hruslov, Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with initial data of step type, Math USSR-Sb. 28 (1976).
[2] A. Cohen and T. Kappeler, Scattering and inverse scattering for step-like potentials, The Schrödinger Equation, 1983, Preprint. · Zbl 0535.34016
[3] Amy Cohen, Solutions of the Korteweg-de Vries equation with steplike initial profile, Comm. Partial Differential Equations 9 (1984), no. 8, 751 – 806. · Zbl 0542.35077 · doi:10.1080/03605308408820347
[4] V. Buslaev and V. Fomin, An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis, Vestnik Leningrad. Univ. 17 (1962), no. 1, 56 – 64 (Russian, with English summary). · Zbl 0243.34013
[5] Stephanos Venakides, The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38 (1985), no. 2, 125 – 155. · Zbl 0571.35095 · doi:10.1002/cpa.3160380202
[6] I, Kay and H. E. Moses, J. Appl. Phys. 27 (1956), 1503-1508.
[7] Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), no. 3, 253 – 290. , https://doi.org/10.1002/cpa.3160360302 Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math. 36 (1983), no. 5, 571 – 593. , https://doi.org/10.1002/cpa.3160360503 Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math. 36 (1983), no. 6, 809 – 829. · Zbl 0527.35074 · doi:10.1002/cpa.3160360606
[8] H. P. McKean, Theta functions, solitons, and singular curves, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 237 – 254.
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